Abstract

Some of the most impressive singular wave fronts seen in Nature are the transbasin oceanic internal waves, which may be observed from a space shuttle, as they propagate and interact with each other. The characteristic feature of these strongly nonlinear waves is that they reconnect whenever any two of them collide transversely. The dynamics of these internal wave fronts is governed by the so-called EPDiff equation, which, in particular, coincides with the dispersionless case of the Camassa-Holm (CH) equation for shallow water in one- and two-dimensions.In this talk, I will present a particle method for the numerical simulation and investigation of solitary wave structures of the EPDiff equation. I will also provide global existence and uniqueness results for this family of fluid transport equations by establishing convergence results for the particle method. The latter is accomplished by using the concept of space-time bounded variation and the associated compactness properties.  Finally, I will present numerical examples that demonstrate the performance of the particle methods in both one and two dimensions. The numerical results illustrate that the particle method has superior features and represent huge computational savings when the initial data of interest lies on a submanifold. The method can also be effectively implemented in straightforward fashion in a parallel computing environment for arbitrary initial data.